6.soru - 1 You are given a ticket for running a red traffic...

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1) You are given a ticket for running a red traffic light. For an observer halted at the red light, the light emits a wavelength λ 0 = 700 nm. You tell the traffic cop that because you were approaching the light, the Doppler shift made it appear green ( λ = 500 nm). How fast would you have been going if this smart-aleck explanation had been true? Please express this speed in units of miles per hour. [Hint: There are 1.6093 kilometers in a mile.] First, Δ λ = λ - λ 0 = 500 nm - 700 nm = - 200 nm . In words, the wavelength decreased by 200 nm, making it appear green (we would call this a blueshift). Δ λ < 0 implies that we’re moving toward the traffic light (as expected). This is a useful consistency check. By convention, if the relative velocity of two objects is negative (i.e., v < 0), then they are getting closer together. This is because the distance that separates them is decreasing as time goes on. Next we determine the driver’s velocity ( v ). The wavelength change is related to velocity by the following formula: Δ λ λ 0 = v c , where Δ λ is the change in wavelength, λ 0 is the wavelength at rest (in the lab), c is the speed of light, and v is radial velocity (along the line of sight). Multiplying both sides of the equation by c , we find: v = c Δ λ λ 0 . Plugging in the numbers from above and c = 3 . 0 × 10 5 km/s, we have: v = (3 . 0 × 10 5 km/s) parenleftbigg - 200 nm 700 nm parenrightbigg = (3 . 0 × 10 5 km/s) × parenleftbigg - 2 7 parenrightbigg = - 8 . 571 × 10 4 km/s Note that the units of wavelength (nm) canceled out. Lastly, we must convert km/s to miles per hour ( mph). The hint indicates that 1 mi = 1 . 6093 km and we know that 1 hr = 3600 s (1 hr = 60 min × 60 s / min = 3600 s). Combining these two we get: 1 km/s = 1 km/s × parenleftbigg 1 mi 1 . 6093 km parenrightbigg × parenleftbigg 3600 s 1hr parenrightbigg 2237 mph Putting all this together we find: v = ( - 8 . 571 × 10 4 km/s ) × parenleftbigg 2237 mph 1 km/s parenrightbigg ≈ - 1 . 92 × 10 8 mph 1
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2) Astronomers doing a ”star census” of the solar neighborhood have found that there are 340 stars within a distance r = 10 pc of the Earth. Compute the volume of a sphere of radius r = 10 pc centered on the Earth. Compute the average number of stars per cubic parsec within this sphere.
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